Key Properties and Applications of Indices in Math
Index is referred to as the power or exponent raised to a number or variable. Index in its plural form is termed as indices. If we write 2³ or a⁵, here 3 and 5 and indices. Each number naturally has an index of 1 but we do not write it as it does not denote any change of value mathematically. If the index is anything other than 1, we require to write it down as the power of the base number. The index of a number can also be zero or negative.
The index represents the number of times a number has to be multiplied by itself. These numbers are governed by several indices rules that we will discuss here. Given below is the representation of the index of a number.
\[a^{n}\] = \[a \ast a \ast a \ast a\cdot \cdot \cdot \cdot\] (n times)
Here,
a is the base and n is termed as the index.
\[2^{4}\] = \[2 \ast 2 \ast 2 \ast 2\] = 16
\[10^{3}\] = \[10 \ast 10 \ast 10\] = 1000
As per indices definition, a number or a variable may have an index. It tells us about how many times the base number is to be multiplied by itself.
Theory of Indices
The laws of indices are a set of fundamental rules that govern the way indexes or indices are to be dealt with mathematically. Indices are not just used to improve the ease of writing the numbers mathematically but also have a specific function and therefore these indices rules are of utmost importance.
Only after knowing these Laws of Indices rules can you solve the algebraic indices problems
We will look at each law of indices formula with index laws examples one by one for various algebraic indices.
Laws of Indices Formulas
Given below are all the laws of indices that you will encounter while dealing with indices. No matter how complex the problem is, these are all the fundamental laws that govern the indices rules.
Multiplication
If two terms with a similar base are to be multiplied by each other, the indices have to be added.
aⁿ . aᵐ = aⁿ⁺ᵐ
Example:
4³ . 4⁶ = 4³⁺⁶ = 4⁹
Division
If two terms with a similar base are to be divided, the indices have to be subtracted
\[\frac{a^{n}}{a^{m}}\] = \[a^{n-m}\]
Example:
\[\frac{5^{6}}{5^{4}}\] = \[5^{6-4}\]
Power of a Power
If the index of a number is itself raised into another power, then the two indices have to be multiplied.
\[(a^{n})^{m}\] = \[(a^{nm})\]
Example:
\[(2^{3})^{4}\] = \[(2^{12})\]
Negative Power
If a term has a negative index it can be represented as reciprocal with the positive index as its power.
\[(a^{-n})\] = \[\frac{1}{a^{n}}\]
Example:
\[(3^{-2})\] = \[\frac{1}{3^{2}}\] = \[\frac{1}{9}\]
Zero Power
If a term has the index as 0, then the value of the term becomes one, no matter what the base value is.
\[a^{0}\] = 1
Example:
\[5^{0}\] = 1
Multiplication with Similar Indices and Different Base
If two terms in multiplication with each other have similar indices but different bases, then the two bases are multiplied with each other.
\[a^{n}\] . \[b^{n}\] = \[(ab)^{n}\]
Example:
\[7^{2}\] . \[5^{2}\] = \[35^{2}\]
Division with Similar Indices and Different Base
If two terms in a division with each other have similar indices but different bases, then the two bases are to be divided with each other.
\[\frac{a^{n}}{b^{n}}\] = \[\left (\frac{a}{b} \right )^{n}\]
Example:
\[4^{2}\] . \[2^{4}\] = \[2^{4}\]
Fractional index
If a term has index in the fraction form it can be represented in the radical form as well.
\[a^{\frac{n}{m}}\] = \[\left ( \sqrt[m]{a} \right )^{n}\]
Example:
\[4^{\frac{2}{3}}\] = \[\left ( \sqrt[3]{4} \right )^{2}\]
You can download the law of indices pdf to revise these index laws examples from time to time in order to be fluent with them.
Laws of Logarithms
Using the Indices rules, we can formulate the laws of indices and logarithms.
Multiplication
\[log_{b}\] (x . y) = \[log_{b}\] (x) + \[log_{b}\] (y)
Example:
\[log_{10}\] (2 . 3) =\[log_{10}\] (2) + \[log_{10}\] (3)
Division
\[log_{b}\] \[\frac{x}{y}\] = \[log_{b}\] (x) - \[log_{b}\] (y)
Example:
\[log_{10}\] \[\frac{2}{3}\] = \[log_{10}\] (2) - \[log_{10}\] (3)
Power of Power
\[log_{b}\] \[x^{m}\] = m. \[log_{b}\] (x)
Example:
\[log_{10}\] \[4^{2}\] = 2 \[log_{10}\] (4)
Zero Power
\[log_{b}\] 1 = 0
1 = \[b^{x}\] , then x=0.
Negative power
\[log_{b}\] \[\frac{1}{x}\] = - \[log_{b}\] (x)
Example:
\[log_{10}\] \[\frac{1}{2}\] = - \[log_{10}\] (2)
Singular Index
\[log_{b}\] = 1
Example:
\[log_{10}\] = 1
Fractional Power
\[log_{b}\] \[\left ( \sqrt[n]{x} \right )\] = \[\left ( \frac{1}{n} \right )\] \[log_{b}\] (x)
Example:
\[log_{10}\] \[\left ( \sqrt[3]{5} \right )\] = \[\left ( \frac{1}{3} \right )\] \[log_{10}\] (5)
FAQs on Laws of Indices: Definition, Rules & Examples
1. What are the 7 laws of indices?
The 7 laws of indices (or exponents) are fundamental rules that help simplify calculations involving powers. These are:
- Product Law: $a^m \times a^n = a^{m+n}$
- Quotient Law: $\frac{a^m}{a^n} = a^{m-n}$
- Power of a Power: $(a^m)^n = a^{mn}$
- Power of a Product: $(ab)^n = a^n b^n$
- Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
- Zero Index Law: $a^0 = 1$, for $a \neq 0$
- Negative Index Law: $a^{-n} = \frac{1}{a^n}$
2. What are the 12 laws of exponents?
The 12 laws of exponents include the standard 7 rules with additional properties and variations:
- Product Law: $a^m \times a^n = a^{m+n}$
- Quotient Law: $\frac{a^m}{a^n} = a^{m-n}$
- Power of a Power: $(a^m)^n = a^{mn}$
- Power of a Product: $(ab)^n = a^n b^n$
- Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
- Zero Exponent Law: $a^0 = 1$
- Negative Exponent Law: $a^{-n} = \frac{1}{a^n}$
- Fractional Exponents: $a^{1/n} = \sqrt[n]{a}$
- One as Exponent: $a^1 = a$
- Exponent of 1: $1^n = 1$
- Multiplication of Exponents with Different Bases but Same Exponent: $a^n \times b^n = (ab)^n$
- Division of Exponents with Different Bases but Same Exponent: $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$
3. How to solve the law of indices?
To solve problems involving the law of indices, follow these steps:
- Identify the appropriate law based on the structure of the expression (for example, if multiplying like bases, use the Product Law).
- Apply the law: Simplify the expression step by step using rules such as $a^m \times a^n = a^{m+n}$ or $(a^m)^n = a^{mn}$.
- Simplify any numbers or variables, keeping track of exponents.
4. What are the 7 laws of exponents with examples?
Here are the 7 laws of exponents along with examples:
- Product Law: $2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$
- Quotient Law: $5^4 \div 5^2 = 5^{4-2} = 5^2 = 25$
- Power of a Power: $(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$
- Power of a Product: $(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000$
- Power of a Quotient: $\left(\frac{4}{2}\right)^2 = \frac{4^2}{2^2} = \frac{16}{4} = 4$
- Zero Index: $7^0 = 1$
- Negative Index: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
5. What is the difference between laws of indices and laws of exponents?
The laws of indices and laws of exponents refer to the same set of rules used to manipulate powers of numbers or variables. They are simply different terms used in different regions or educational systems, but their mathematical principles are identical. Vedantu explains both terms clearly, ensuring that students understand the concepts regardless of terminology.
6. Why are the laws of indices important in mathematics?
The laws of indices are crucial in mathematics because they:
- Simplify complex algebraic expressions
- Make calculations with large or small numbers more manageable
- Lay the foundation for advanced topics such as logarithms, scientific notation, and polynomial operations
7. How can I practice laws of indices with real-life examples?
You can practice the laws of indices through real-life applications such as:
- Calculating compound interest, where exponents represent time periods
- Measuring area and volume, which often involve powers
- Understanding scientific notation for expressing very large or small values
8. How do negative and fractional indices work?
A negative index means taking the reciprocal of the base: $a^{-n} = \frac{1}{a^n}$. A fractional index signifies a root: $a^{1/n} = \sqrt[n]{a}$ and more generally $a^{m/n} = \sqrt[n]{a^m}$. These laws allow exponents to represent roots and reciprocals, and Vedantu explains their applications thoroughly in video lessons and tutorial sessions.
9. What are common mistakes students make when applying the laws of indices?
Common mistakes include:
- Incorrectly adding instead of multiplying exponents in expressions like $(a^m)^n$
- Mixing up product and quotient laws
- Misplacing negative or zero exponents
- Confusing different bases as the same
10. How do the laws of indices help in algebraic simplification?
The laws of indices enable rapid simplification of algebraic expressions involving powers by applying systematic rules. For example, $x^2 \times x^3 = x^{2+3} = x^5$. These rules reduce complex expressions to a simpler form quickly, boosting both speed and accuracy in exams. Vedantu’s expertly designed practice problems let students hone these skills regularly.
